First, coercion is not in any way essential for polynomial rings. It makes some things a bit more cumbersome, but it is certainly workable by putting elements in the tropical semiring.
Second, the operations between, e.g., integers and tropical ring elements does *not* make sense. What is T(1) + 1? Is it 1 or 2? There's no good way to answer this. The case with (integer) 0 under addition is a special case that I would like to not have, but changing that would be more work than the benefit. Best, Travis On Sunday, March 31, 2024 at 6:45:12 AM UTC+9 anime...@iiits.in wrote: > I read about Matrix Algebra over Tropical Semiring and found that there is > "tropical determinant" for this purpose (Page 3 : Eq 4 > <http://www.math.uni-konstanz.de/~michalek/may22.pdf>). This formulation > does not use (additive) inversion. > > I was also trying different operations between tropical elements and other > elements(like Integers). It looks like there is a need for implementation > of coercion maps between few datatypes and tropical elements. > One can check this by running few commands like > sage: T = TropicalSemiring(QQ) > sage: a = T(1); b = T(0) > sage: a + 0; b + 0 > sage: a * 0; b * 0 > sage: a + 1; b + 1 > sage: a * 1; b * 1 > > Coercion maps and models are crucial for polynomial element implementation. > For example : between > 'Symbolic Ring' and 'Tropical semiring over Rational Field' > 'Univariate Polynomial Ring in x over Tropical semiring over Rational > Field' and 'Tropical semiring over Rational Field' > 'Tropical semiring over Rational Field' and 'Integer Ring' > On Friday, March 29, 2024 at 12:16:44 AM UTC+5:30 tcscrims wrote: > >> One needs to be *very* careful about what operations mean. -T(2), the >> (tropical) additive inverse of 2, is *not* T(-2). There is no way to >> have min(2, a) = \infty (note that this is the (tropical) additive unit). >> >> I am not sure how (or if) determinants over the tropical semiring are >> defined, but one could define it by having the sign included in each term. >> So for the 2x2 case, it could be T(a*b) + T(-b*d), but I don’t know if this >> makes det a group homomorphism. >> >> Best, >> Travis >> >> >> On Wednesday, March 27, 2024 at 2:17:39 AM UTC+9 anime...@iiits.in wrote: >> >>> This problem can be seen clearly if we use *Matrix_generic_dense* as >>> element class for matrix space. >>> >>> >>> On Tuesday, March 26, 2024 at 9:42:45 PM UTC+5:30 Animesh Shree wrote: >>> >>>> I was going through this code and got error. But I could not understand >>>> why this is the case. >>>> >>>> >>>> >>>> >>>> sage: T = TropicalSemiring(QQ) >>>> sage: T(1) >>>> 1 >>>> sage: T(2) >>>> 2 >>>> sage: T(-2) >>>> -2 >>>> sage: -T(2) >>>> >>>> --------------------------------------------------------------------------- >>>> ArithmeticError Traceback (most recent call >>>> last) >>>> Cell In[26], line 1 >>>> ----> 1 -T(Integer(2)) >>>> >>>> File ~/sage/src/sage/rings/semirings/tropical_semiring.pyx:278, in >>>> sage.rings.semirings.tropical_semiring.TropicalSemiringElement.__neg__() >>>> 276 if self._val is None: >>>> 277 return self >>>> --> 278 raise ArithmeticError("cannot negate any non-infinite >>>> element") >>>> 279 >>>> 280 cpdef _mul_(left, right) noexcept: >>>> >>>> ArithmeticError: cannot negate any non-infinite element >>>> sage: >>>> >>>> >>>> It looks like starting with T(-2) and reaching to -2 from T(2) by >>>> comparing with zero(+Inf) are different things. >>>> T(-2) = -2 >>>> T(2) -T(2) = T.zero(+inf) = T(2) + (-T(2)) >>>> >>>> My doubt is : if we cannot negate the elements, then how can we >>>> compute the determinant of a Matrix over Tropical Semiring. >>>> For example in 2x2 matrix : [[a,b], [c,d]] the determinant should be ad >>>> - bc >>>> But can it be expressed as ad + (-bc) -->> min ( add(a,d), >>>> -1*add(b,c) )? >>>> >>>> In fact we connot even do matrix subtraction directly. >>>> What can be done in these cases?? >>>> On Thursday, March 21, 2024 at 8:48:26 PM UTC+5:30 Animesh Shree wrote: >>>> >>>>> Hello Sir, >>>>> >>>>> I have submitted my proposal. >>>>> Please review it and let me know necessary updates and improvements. >>>>> >>>>> I want to verify soundness of my approach and extend the proposal for >>>>> Multivariate Polynomials. >>>>> >>>>> Thank You >>>>> Animesh Shree >>>>> On Tuesday, March 12, 2024 at 12:26:38 AM UTC+5:30 tcscrims wrote: >>>>> >>>>>> Mathematically speaking, you can always weaken axioms. However, there >>>>>> are some extra advantages that additive groups have that commutative >>>>>> semirings don't have (mainly 0, the additive identity). >>>>>> >>>>>> That being said, there isn't anything prevent you from constructing >>>>>> the appropriate categories. It would be good to have a more specific >>>>>> use-case in mind, but that isn't necessary. However, one should be >>>>>> careful >>>>>> with the name because it would conflict with what "most" people would >>>>>> call >>>>>> an algebra (which is why we have MagmaticAlgebras). >>>>>> >>>>>> Best, >>>>>> Travis >>>>>> >>>>>> >>>>>> On Tuesday, March 12, 2024 at 2:57:29 AM UTC+9 anime...@iiits.in >>>>>> wrote: >>>>>> >>>>>>> Hello Sir, >>>>>>> >>>>>>> I was going through "Algebras" and I had a doubt. >>>>>>> Does MagmaticAlgebra and AssociativeAlgebra have to be implemented >>>>>>> over Ring only? >>>>>>> I went through internet and google uses rings to define those >>>>>>> algebras, but the axioms that those algebra follow (Unital, >>>>>>> Associative) >>>>>>> are also preserved by commutative semirings. >>>>>>> Would it be good to define MagmaticAlgebra and AssociativeAlgebra >>>>>>> over other objects that follow those axioms too or come-up with >>>>>>> alternative >>>>>>> Algebra? >>>>>>> >>>>>> -- You received this message because you are subscribed to the Google Groups "sage-gsoc" group. 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